MPA Lectures onGravitational Waves in Cosmology

Azadeh Maleknejad

Max-Planck-Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany

Abstract

Almost a century ago, Albert Einstein predicted the existence of gravitational waves,ripples in the spacetime, as a possible solution of the linearized general relativity. Surpris-ingly, shortly after that, Einstein himself changed his mind, and since after, like manyother physicists of his time, he believed that they are not physical but an artifact oflinearization. It was not until Herman Bondi’s 1957 Nature paper that the first mathe-matically precise definition of gravitational waves in the full Einstein equations has beendiscovered. Seventeen years later, Hulse and Taylor made the first indirect detectionof this mysterious waves. And finally, in another historic event, only three years ago,these waves were first detected directly by LIGO. Apart from the astrophysical gravita-tional waves, a stochastic background of relic gravitational waves is also highly expected.These tensor perturbations are the only missing (key) prediction of the inflation paradigmwhich has not been detected yet. These primordial gravitational waves, either vacuum orsourced by relic particles, are free-streaming since inflation and can teach us a lot aboutthe physics at its highest possible energy scales. This talk has two parts. In the first part,I will talk about the gravitational waves in asymptotically flat spacetimes and will ex-plain the Bondi’s brilliant formalism to prove the physicality of the gravity waves. In thesecond part, I focus on the gravitational waves in our expanding cosmological universe.I explain the spin-2 fluctuations generated during inflation and the resulting stochasticgravitational background.

Contents

I Asymptotically flat spacetimes 1

1 Linearized GR and Gravitational Waves 11.1 The weak-field metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 General solution of the linearized field equation . . . . . . . . . . . . . . . . . 21.3 The compact-source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Polarization states and effect on free particles . . . . . . . . . . . . . . . . . . 5

2 Physicality of Gravitational radiation?! 82.0.1 Theoretical issues of linear gravity . . . . . . . . . . . . . . . . . . . . 8

2.1 Geometric definition for GWs . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Petrov classification for Weyl tensor . . . . . . . . . . . . . . . . . . . . . . . 92.3 Gravitational radiation after Bondi . . . . . . . . . . . . . . . . . . . . . . . . 11

II Expanding Universe 13

3 Gravitational Waves in expanding Universe 133.1 Cosmological background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Cosmic inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Gravitational waves in cosmological eras 164.1 Inflation and primordial gravitational waves . . . . . . . . . . . . . . . . . . . 18

4.1.1 Adiabatic perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 (Almost) scale invariant Power spectrum . . . . . . . . . . . . . . . . . . . . . 204.3 Nearly non-Gaussian stochastic field . . . . . . . . . . . . . . . . . . . . . . . 21

5 Adiabatic modes and inflationary consistency relations 235.0.1 Weinberg’s Adiabatic modes . . . . . . . . . . . . . . . . . . . . . . . 235.0.2 Gaussianity and Maldacena’s consistency relation . . . . . . . . . . . 25

6 Gravitational waves and CMB 266.1 Integrated SachsWolfe effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 CMB polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References 30

Part I

Asymptotically flat spacetimes

1 Linearized GR and Gravitational Waves

Shortly after proposing general relativity, Albert Einstein linearized his field equation

Rµν −1

2Rgµν = 8πGTµν , (1)

1

and realized that general relativity admits solutions in which the fluctuations of the Minkowskispace-time are plane waves traveling with the speed of light. In the following, we explainsome basic features of these linear solutions of general relativity.

1.1 The weak-field metric

A weak gravitational field corresponds to a region of the spacetime that is weakly curved. Inother words, throughout such region, there exist coordinate systems in which the spacetimemetric takes the form

gµν = ηµν + hµν , (2)

in which hµν and all its partial derivatives are small

hµν 1 and ∂nλ hµν 1 (n > 1). (3)

Note that we can well consider small perturbations about some other background metric, suchthat gµν = gµν + hµν . In particular, we will use a similar weak field approach for cosmologicalperturbations around FRW metric in section 3.

In the weak-field general relativity, we expand the field equations in powers of hµν andkeep the linear terms. In fact, from the Einstein equation in (1), we find the linearizedgravitational field equation for hµν

∂α∂αhµν + ∂µ∂ν h− ∂ν∂λhλµ − ∂µ∂λhλν − ηµν(∂α∂

αh− ∂σ∂λhλσ) = −16πGTµν , (4)

where h is the trace of the field, h = hµµ. Although linearized, the above equation does notlook like a wave-equation. However, this messy equation can be simplified in terms of thefield redefinition

hµν ≡ hµν −1

2ηµν h, (5)

which is the trace-reverse cousin of h, i.e. h = −h. Moreover, these fields are defined only upto gauge transformations, and in this case, the most convenient gauge is the Lorenz gauge

∂µhµν = 0. (6)

The linearized field equation in terms of hµν in the Lorenz gauge takes the most simple formbelow

∂α∂αhµν = −16πG Tµν . (7)

which in the vacuum has the desired wave-like form.

1.2 General solution of the linearized field equation

The linearized equation in vacuum has plane wave solutions of the form

hµν = Aµν(kλ) exp(ikλxλ), (8)

whereAµν are constant components of a symmetric tensor and kµ is the wave vector. Equation(7) implies that the wave vector is null, kµk

µ = 0. Therefore, the homogeneous linear Einsteingravity has the following general solution

hµν =

∫d3k

(Aµν(~k) exp(ikµx

µ) +A∗µν(~k) exp(−ikµxµ)

), (9)

2

which is the superposition of all possible plane waves.In the presence of a source with nonzero Tµν , we need to solve the inhomogeneous equation

with the Green’s equation

∂α∂αG(xµ − yµ) = δ4(xµ − yµ), (10)

where the required retarded Green’s function for x0 > y0 is

G(xµ − yµ) =1

(4π)|~x− ~y|δ(x0 − y0 − |~x− ~y|). (11)

Therefore, the sourced part of the linear gravitational equation is

hµν(~x, t) = −4G

c4

∫d3y

Tµν(~y, ct− |~x− ~y|)|~x− ~y|

. (12)

In fact, the gravitational field at an event point (t, ~x) is the integral over the past lightconeof the event point occupied by the source.

Figure 1: The change in the gravitational field at an event point (t, ~x) is the sum of the effects of the source’sTµν at the point ((t− |~x− ~y|), ~y) on the past lightcone.

1.3 The compact-source

For astrophysical purposes, the gravitational source has a spatial size much smaller than thedistance to the point of the observation. In such cases, it is sufficient to consider the firstterm in the Taylor expansion

1

|~x− ~y|=

1

r+yixir3

+ yiyj(

3xixj − r2δijr5

)+ ..., (13)

3

where r = |~x| is the spatial distance from the origin to the field point. In particular, thelinear solution (12) has the far-field approximation

hµν(~x, t) = −4G

c4r

∫d3yTµν(~y, ct− r), (14)

which decays as 1/r. The physical meaning of each component in the above is as follows:

•∫T 00d3y = Mc2 is the total energy of source particles,

•∫T 0id3y = P ic is the total momentum of source particles in the xi-direction,

•∫T ijd3y = Πij is the integrated internal stresses in the source.

For an isolated source, the conservation of energy-momentum tensor reads

∂µTµν = 0.

Therefore, in the linear theory, M and P i are conserved. Furthermore, for a compact object,P i can be written as

P i =∂

∂y0

[ ∫T 00(y0, ~y)yid3y

]y0=ct−r

, (15)

which can always set to zero by choosing the origin of the coordinate system at the source’scenter of mass. Therefore, we can always work in a coordinate system associated to the centerof momentum frame of the source particles in which P i = 0.

Finally, in the center of momentum coordinate, we have the desired hµν components as

h00 = −4GM

c2r, (16)

h0i = 0, (17)

hij = −4G

c4r

∫Tij(ct− r, ~y)d3y. (18)

It is possible to further simplify the form of hij for the compact-source case. We can write 1∫T ijd3y = −1

2

∫ (∂k(T

ik)yj + ∂k(Tjk)yi

)d3y. (19)

Moreover, the energy-momentum conservation leads to∫∂k(T

ik)yjd3y = − d

dy0

[ ∫T i0yjd3y

], (20)

By using the energy-momentum conservation once again and dropping the total derivativeterm, we have ∫ (

T i0yj + T j0yi)d3y =

d

dy0

[ ∫T 00yiyjd3y

]. (21)

Thus, in terms of the quadruple-moment tensor of the source

Iij(y0) =

∫T00(y0, ~y)yiyjd3y, (22)

1Note that enclosing the integral outside the source, we dropped the total derivative term in the RHS,∫∂k(T ikyj)d3y = 0.

4

we can express hij in the far-field limit as

hij(t, ~x) = −2G

c6r

d2

dy02

[Iij(y

0)

]y0=ct−r

. (23)

The above can be decomposed into the trace and a traceless tensor, γij , as

hij(t, ~x) = hδij + γij , (24)

where γij is

γij = −2G

c6r

d2

dy02

[Jij(y

0)

]y0=ct−r

, (25)

and Jij is the reduced quadrupole-moment tensor of the source distribution

Jij = Iij −1

3δijI

kk . (26)

1.4 Polarization states and effect on free particles

In the previous section, we saw that similar to the Maxwell’s equation which predicts elec-tromagnetic waves, the linearized GR also suggests the existence of gravitational waves. Inthis section, we discuss the propagation of gravitational radiation in flat space.

Let us consider a general gravitational perturbation satisfying the empty-space linearizedfield equation and the Lorenz gauge condition (see (9)). Under a coordinate (gauge) trans-formation of the form

xµ 7→ x′µ = xµ + ξµ, (27)

the transformed perturbation

h′µν = hµν − ∂µξν − ∂νξµ, (28)

is still in the Lorenz gauge condition provided that the 4-vector ξµ satisfies ∂ν∂νξµ = 0. Also

the trace-reverse field tensor in (5) transforms as

h′µν = hµν − ∂µξν − ∂νξµ + ηµν∂αξα. (29)

We can use the above gauge transformation to set any four linear combinations of hµν tozero. In particular, the transverse-traceless gauge, so called TT gauge, is defined by choosing

hTT0i = 0 and hTT = 0. (30)

Moreover, we are still in the Lorenz gauge which in the TT gauge gives

∂0hTT00 = 0 and ∂ihTTij = 0. (31)

Notice that for non-stationary (time-dependent) gravitational fields, as for a general gravi-tational wave disturbance, it implies that hTT00 also vanishes. As a result, for those cases, weare left with the spatial components of a symmetric, traceless and transverse tensor, i.e. wehave only two dynamical degrees of freedom.

5

For a given gravitational wave with a spatial wave-vector ~k = kn in an arbitrary coordinatesystem, we can read the form of the wave in the TT gauge as

ATTij = (P ki Plj −

1

2P klPij)Akl, (32)

where Pij is the spatial projection tensor

Pij = δij − ninj , (33)

which projects tensors to the 2 dimensional surface normal to n. Similar to the electro-magnetic waves, the two dynamical modes in the ATTij can be decomposed in terms of twopolarization states. Writing n in terms of polar and azimuthal angles, θ and φ, as

n = (sin θ cosφ, sin θ sinφ, cos θ), (34)

one can define the complex polarization vectors

e±(n) =1√2

(θ ± iφ), (35)

where θ and φ are orthogonal unit vectors in the plane perpendicular to n, in the directionsof increasing θ and φ, respectively. In terms of e± and (θ, φ), we can construct two types ofpolarization states:

i) the linear, plus and cross, polarization tensors,

ePlusij (n) ≡ 1√

2

(θiθj − φiφj

)and eCross

ij (n) ≡ 1√2

(θiφj + φiθj

), (36)

and ii) the circular, right- and left-handed, polarization tensors

eR,Lij (n) ≡ e±i e±j . (37)

Notice that these two pair of tensors are related as

eR,L(n) =1√2

(ePlus ± ieCross

)(38)

For n = x3, the P=plus and C=cross polarization tensors can be written in terms of thecomponents

eP11 = −eP22 = eC12 = eC21 =1√2, and eP12 = eP21 = eC11 = eC22 = eP,Ci3 = 0, (39)

while the circular ones are given as

eR,L11 = −eR,L22 =1√2, eR,L12 = eR,L21 = ± i√

2and eR,L3i = eR,Li3 = 0. (40)

In order to have a feeling about the effect of each of the above polarization states ofgravitational waves, it is useful to consider their effect on the geodesic deviation of free-falling particles. Consider two nearby (non-interacting) free-falling particles initially at restwith separation vector

ξµ0 = (0, ξi0). (41)

6

Figure 2: Effect of gravitational wave in different polarization states on a ring of freely-falling particles. Thecontinuous lines and the dark filled dots show the positions of the particles at different times, while the dashedlines and the open dots show the unperturbed positions. This illustration is barrowed from [1].

The arrival of a gravitational wave will perturb the geodesic motion of the two particles andproduce a nonzero contribution to the geodesic-deviation equation. We recall the geodesic-deviation equation, the changes in the separation four-vector Xµ between two geodesic tra-jectories with tangent four-vector uµ, is

D2Xµ

Dτ2= −Rµνλσu

νuλXσ, (42)

whereD

Dτ≡ uµ∇µ,

is the covariant time derivative along the geodesic of a particle. Therefore, the geodesicdeviation of the nearly particles in the presence of the GW is

D2Xi

Dτ2= −Ri0j0Xj . (43)

In the rest frame of particle A, around the particle the connection vanishes and in the TTgauge the coordinate time and the particle’s proper time coincides at the leading order (t =τ +O((hTTµν )2)), and we simply have

d2Xi(t)

dt2=

1

2∂2t h

TTij ξ

j , (44)

7

which gives the separation vector as

Xi(t) = ξj(δij +

1

2hTTij (t)

). (45)

In figure 2, we present this effect by each of the 4 possible polarization states of GWs on aring of freely-falling particles.

2 Physicality of Gravitational radiation?!

Shortly after proposing general relativity, Albert Einstein linearized it and predicted theexistence of gravitational waves as a possible solution of his theory. However, he mistakenlythought they are not the solutions of the full nonlinear theory and therefore unphysical. Headvanced arguments against the existence of gravity waves, which stopped the development ofthe subject for decades. It was only after his death that the actual physical nature of GWs wasunderstood. In this section, first, I briefly mention the ambiguity and fundamental problemswhich led to this mistake. Then, I will explain the mathematically and geometrically precisedefinition of GWs by H. Bondi, F. Pirani, I. Robinson, and A. Trautman which proved theexistence of GWs as the solutions of the full theory.

2.0.1 Theoretical issues of linear gravity

Unlike Maxwell’s equations, gravitation is a non-linear theory. That is the direct result ofthe fact that gravity gravitates. More precisely, any energy-momentum acts as a sourceof gravity, including its energy-momentum tensor. That is unlike the electromagnetic field(photons in QED) which are not charged. So an important question that one should addressbefore taking the linear solutions too seriously is:

Do the fully nonlinear Einstein equations admit solutions that can be described as grav-itational waves? If yes, are they coincide with the linear solutions far from the source? Ifeverything is fine, then we need to solve some other subproblems, including;

i) What is the definition of a plane gravitational wave in the full theory?ii) Do they carry energy and angular momentum? Etc.In 1958, H. Bondi formulated the first mathematically precise definition of GWs, which

followed by papers by H. Bondi, F. Pirani, I. Robinson, and A. Trautman proved the existenceof GWs as the solutions of the full theory and confirmed that they do carry energy. Here,I briefly review their stunning results. For a nice and recent review on the mathematics ofGWs and the history behind it, see [2]. More details about the asymptotically flat spacetimes,BMS group, and the IR gravity can be found in [3].

2.1 Geometric definition for GWs

Even before Bondi’s Nature paper, Felix Pirani used a Brilliant idea as the first attempt ata purely geometric definition of spacetime with gravitational waves. Pirani’s intuition wasbased on using the algebraic form of the Weyl tensor, the so-called Petrov classification tosee if spacetime has radiative regions. That intuition makes sense since far from the sourcethe Ricci tensor vanishes, and the Riemann tensor reduces to Weyl. Here we briefly discussthis idea and the resulting geometrical description of radiative spacetimes.

In 4d, the Riemann curvature tensor has 20 independent components in which 10 indepen-dent components are associated to the Ricci tensor. The other 10 independent components

8

can be assigned to the (traceless) Weyl tensor which is a conformal tensor

Cµνλσ = Rµνλσ − (gµ[λRσ]ν − gν[λRσ]µ) +1

3R gµ[λgσ]ν . (46)

More precisely, it is unchanged under a conformal transformation of the metric 2

gµν 7→ g′µν = Ω2gµν , (47)

C σµνλ 7→ C

′ σµνλ = C σ

µνλ . (48)

More intuitively, the Weyl tensor expresses the tidal forces that a free-falling body feels alonga geodesic. However, unlike the Ricci tensor, it does not have information about the change ofthe volume, but only how the shape of the body is distorted by the tidal forces (see illustrationin figure 3).

Figure 3: Weyl vs. Ricci. Weyl tensor is blind to the scaling and change of volume while Ricci is blind to thetidal forces!

2.2 Petrov classification for Weyl tensor

The Petrov classification describes the possible algebraic symmetries of the Weyl tensor ateach event in a Lorentzian manifold. This classification was found in 1954 by A. Z. Petrovand independently by Felix Pirani in 1957. The classification is based on the observationthat an arbitrary asymmetric tensor

Xµν = −Xνµ,

under the action of the Weyl tensor transforms to another asymmetric tensor as

XµνCµναβ = Yαβ. (49)

Therefore, the natural question is to find the eigen-bivectors and eigenvalues of the aboveequation, i.e.

1

2XµνC

µναβ = λXαβ. (50)

2Therefore, we have C′µνλσ = Ω2Cµνλσ.

9

The Weyl tensor can have at most four linearly independent eigen-bivectors at each givenevent which are associated with some null vectors in the original spacetime, called the princi-pal null directions (PND). The Petrov classification states that there are precisely six possibletypes of algebraic symmetries, known as the Petrov types:

• Type I: four simple PNDs,

• Type II: two simple PNDs and two PNDs coincide,

• Type D: two pairs of coinciding PNDs,

• Type III: one triple and one simple PNDs,

• Type N: all PNDs coincide,

• Type O: the Weyl tensor vanishes.

We present these six types of the Weyl tensor in figure 4. Let us take a closer look at thePetrov types D, N, and O.

Figure 4: The schematic form of the Petrov classification of the Weyl tensor. The green arrows correspond toeach of the principal null directions and the parallel green arrows represent the number of coinciding PNDs.In the O type, the Weyl tensor vanishes.

Type D regions are associated with the gravitational fields of isolated massive objects, e.g.,stars and black holes, which is entirely characterized by its mass and angular momentum.The two coincided PNDs present radially ingoing and outgoing null congruences near theobject.

Type O regions are conformally flat places with zero Weyl tensor, e.g., exact Minkowskiand FRW. In this case, any gravitational effects must be due to the immediate presence ofmatter or the field energy of some non-gravitational field.

Type N regions are those regions with transverse gravitational radiation. A spacetimeregion is type N, if and only if there exists a null vector, kµ, such that

Cµνλσkσ = 0. (51)

The four coinciding PNDs are given by this null vector which is the wave vector of thepropagating gravitational wave.

It is noteworthy to mention that the Petrov type of a spacetime may vary from regionto region. For instance, in figure 4, the blue arrow shows the direction of change of Petrov

10

type of Weyl tensor as we approach null infinity in an asymptotically flat spacetime. 3 Tosummarize, the Weyl tensor of a radiative spacetime must be of type N very far from thesources, i.e. in the asymptotic future.

2.3 Gravitational radiation after Bondi

Herman Bondi in his Nature paper [5] followed by a subsequent paper by Bondi, Pirani andRobinson [6], provided the first mathematically precise definition of gravitational waves inthe full Einstein equation. Moreover, he proved that gravitational radiation carries energy.Here, we review that in asymptotically flat spacetimes. In this part, we adopt the Bondi’s(u, r, z, z) coordinate largely because they are used in most of the literature on asymptoticallyflat spacetimes.

Figure 5: The Penrose diagrame of an asymptotically flat spacetime. The future/past null infinities, I±, areparametrized by retarded/advanced Bondi time u/v. The red ring, i0, is the spatial infinity. The two bluecones in the upper half, present two null surfaces specified with u = u1 and u = u2. The shaded area is thecut in I+ with u1 < u < u2.

Far from the source, the spacetime is very close to flat Minkowski space

ds2 = −du2 − 2dudr + 2r2ηzzdzdz, (53)

and throughout we use the Bondi retarded coordinate (u, r, z, z) which in terms of the spher-ical coordinate (t, r, θ, φ) are given as

u = t− r, z = cotθ

2eiφ and z = cot

θ

2e−iφ. (54)

3This effect is due to the peeling theorem in general relativity which describes the asymptotic behavior ofthe Weyl tensor as one goes to null infinity. Let γ be a null geodesic from a point p to null infinity, with affineparameter λ. Then the theorem states that, as λ approaches infinity

Cabcd =C

(1)abcd

λ+C

(2)abcd

λ2+C

(3)abcd

λ3+C

(4)abcd

λ4+ ..., (52)

where C(1)abcd is type N, C

(2)abcd is type III, C

(3)abcd is type II and C

(4)abcd is type I [4].

11

Here u is the Bondi (retarded) time while the advanced time (in Minkowski) is

v = t+ r. (55)

• Definition I. Asymptotic flatness: An asymptotically flat spacetime is a Lorentzian man-ifold in which, the curvature vanishes at large distances from some region so that thegeometry becomes indistinguishable from Minkowski. Outside the source, the Riccitensor is zero and therefore, the asymptotic flatness impose some asymptomatic falloffconditions on the Weyl tensor.

• Definition II. Future null infinity: I+ is defined as endpoints of all future-directed nullgeodesics along which r →∞. This null surface is the product of S2 with a null line utaking values in R. Each null hypersurface, σu0 , intersects I+ in a 2-sphere with u = u0.(See figure 5)

Now, we want to study gravitational theories in which the metric is asymptotic, but notexactly equal to, the flat metric, and we abbreviate ΘA = (z, z). Choosing the Bondi gauge

grr = grA = 0, (56)

∂r det(gABr2

)= 0, (57)

the most general four-dimensional metric has the form

ds2 = −Udu2 − 2e2βdudr + gAB

(dΘA +

1

2UAdu

)(dΘB +

1

2UBdu

). (58)

Notice that the gauge condition (56) completely fixed the local diffeomorphisms. Moreover,it implies that r is the luminosity distance. Up to now, we just wrote a general metric inthe specific gauge (56). Any geometry can be described locally by the metric (58). Imposingthe asymptotic flatness condition at large r with fixed (u, z, z) leads to falloff conditions onthe metric components. For the natural choice made by Bondi, van der Burg, Metzner, andSachs (BMS) [7], the large-r structure of the metric is constrained to be

ds2 = −du2 − 2dudr + 2r2ηzzdzdz

+2mB

rdu2 + rγzzdz

2 + rCzzdz2 +Dzγzzdudz +Dzγzzdudz

+1

r

(4

3(Nz + u∂zmB)− 1

4∂z(γzzγ

zz)

)dudz + c.c.+ . . . ,

(59)

where Dz is the covariant derivative with respect to ηzz while γzz, mB and Nz are r indepen-dent and functions of (u, z, z). The first three terms in (59) are simply the flat Minkowskimetric, and the remaining terms are the leading corrections:

• The first quantity mB(u, z, z) is the Bondi mass aspect (for Kerr BH, mB = GM),

• The next one is Nz(u, z, z) which is the Bondi angular momentum aspect (for a KerrBH Nz = 2GmL),

• The last term is γzz which describes the gravitational waves. This quantity is transverseto I+ and r−1-suppressed comparing to the dominant orders. The Bondi news tensor isdefined as

Nzz = ∂uγzz, (60)

which is the gravitational analogue of the field strength in gauge field theories, i.e.Fuz = ∂uAz.

12

• Definition III. The Bondi mass at a Bondi time, u1, is defined as the integral over S2

(the sphere with u1 at I+), as

MB(u1) =1

4πG

∫S2

d2zηzzmB(u1, z, z). (61)

The Bondi mass is positive and time dependent, such that it is always non-increasing withtime. Moreover, in the limit u → −∞, S2

u asymptotically approaches the spatial infinity, i0

and the Bondi mass is equal to the (conserved) ADM mass [8, 9]

MADM = limu→−∞

MB(u). (62)

Moreover, the time evolution of mB is given by the Einstein equation component Guu at I+

as

∂umB =1

4

[D2zN

zz +D2zN

zz]− 1

4NzzN

zz − 4πG limr→∞

[r2TMuu

], (63)

where TMuu is the matter field’s energy-momentum tensor. Using the Einstein equation in (61)and considering a compact source with r2TMuu ∼ O(r−1) at future null infinity, we have

MB(u2)−MB(u1) = −1

4

∫ u2

u1

du

∫d2zηzzNzzN

zz, (u2 > u1), (64)

in which the D2zN

zz terms vanishes under the S2 integral. This is the famous Bondi mass-loss formula which measures the amount of mass-loss after some radiation through I+ (seefigure 5). That is zero in case that the Bondi news vanishes. Otherwise, the Bondi mass isdecreasing with time in the form of gravitational radiation.

The angular momentum aspect,Nz(u, z, z),

defined in the above is governed by a constraint equation

∂uNz =1

4∂z(D

2zC

zz −D2zC

zz)− u∂u∂zmB −1

4∂z(CzzN

zz)− 1

2CzzDzN

zz + 8πG limr→∞

r2TMuz .(65)

Integrating the above similar to the mass aspect, one can find the amount of angular mo-mentum carried by the GWs.

Part II

Expanding Universe

3 Gravitational Waves in expanding Universe

In the first part, we studied gravitational radiation in asymptotically flat spacetimes. Inthis part, we want to explore the generation and evolution of gravitational waves in thecosmological background.

13

Cosmological era Eq. of state Scale factor Hubble SEC

Cosmic inflation w ' −1 a(t) = Exp(Ht) H(t) = Hinf No

(ε = − HH 1) w = −1 + 2

3ε a(τ) = − 1Hτ H(τ) = − 1

τ

Radiation era w = 13 a(t) =

(ttI

) 12 H(t) = 1

2t Yes

a(τ) = ττI

H(τ) = 1τ

Matter era w = 0 a(t) =(ttJ

) 23 H(t) = 1

3t Yes

a(τ) =(ττJ

)2 H(τ) = 2τ

Table 1: Equation of state, scale factor, and the Hubble parameter for the each cosmological eras. The lastcolumn shows the validity or violation of strong energy condition (SEC) during that era, i.e. if ρ + 3P > 0.Here H ≡ aH.

3.1 Cosmological background

Modern cosmology is based on two key observational facts: i) the universe is expanding andii) on large scales (> 100Mpc) the matter distribution is homogeneous and isotropic. Theaverage spacetime is then described by the Friedmann-Lemaıtre-Robertson-Walker (FLRW)metric

ds2FLRW = −dt2 + a2(t)(

dr2

1− kr2+ r2dΩ2), (66)

where t is the cosmic time, a(t) is the scale factor, and k = 0,+1,−1 describe flat, positivelycurved and negatively curved spacelike 3-hypersurfaces, respectively. From now on, we re-strict our discussion to the case of the flat universe with k = 0 which is favored by presentobservations.

For a perfect fluid with energy density, ρ, and equation of state, w

P = wρ,

we have the Friedmann equations as

3M2Pl(a

a)2 = ρ and M2

Pl

a

a= −1

6(1 + 3w)ρ. (67)

The scale factor is a function of only time and during different cosmological eras is given inthe table 1.

The cosmic time, t, is related to the conformal times, τ , as

τ ≡∫

dt

a(t). (68)

Solving for the scale factor in (67), we have a(t) in terms of the cosmic time and conformaltime respectively as

a(t) =

(t

tI

) 2(1+3w)+2

and a(τ) =

(τ

τI

) 2(1+3w)

, (69)

where tI and τI are some positive constants.

14

τ

τ0

τrec

τi r

BA

θ

Figure 6: The (comoving) causal past of an observer today at τ0 (redshift z0 = 0), in FLRW spacetime madeof ONLY ordinery matter, i.e. 1 + 3w > 0. The orange line shows the last scattering surface, τrec (at redshiftzrec ' 1090). A and B are two causaly disconnected points at last scattering surface. In fact, the angle spanedby the shaded area at the last scattering surface is the comoving horizon at recombination.

In order to understand the causal structure of the cosmological spacetime, ds2 = 0, weconsider the radial null geodesics given as

r(τ)− r(τI) =

∫ τ

τI

dτ ′ =

∫ a

aI

d ln a′

a′H(a′), (70)

which is specified in terms of the comoving (particle) horizon, (aH)−1, as

(aH)−1 =(1 + 3w)

2τI a

(1+3w)/2. (71)

Horizon problem:Ordinary forms of matter, with positive pressure, satisfy the strong energy condition

(SEC), i.e. ρ + 3P > 0. Thus from (71), the comoving horizon increases as the universeexpands. Now, let us compute the angle spanned by the comoving horizon at recombination,θ. As we see in figure 6, θ is given as

sin θ =2(τrec − τi)τ0 − τrec

. (72)

On the other hand, we can read the τ integral in (68) as

τ − τI = −∫ z2

z1

dz

H(z), (73)

15

where z is the redshift parameter

1 + z =1

a(z), (74)

in which we set the scale factor today to unity, a0 = 1. Moreover, the Hubble parameter is

H(z) =√

Ωm(1 + z)3 + Ωγ(1 + z)4 + ΩΛ, (75)

where Ωm = 0.3, Ωγ = Ωm1+zeq

, and ΩΛ = 1 − Ωm are the matter, radiation, and (late time)

dark energy fraction (zeq = 3400). Using (73) and (75) in (72) and solving the integral, wefind

θc ' 2.3. (76)

Therefore, causal theories should have vanishing correlation functions for θ > θc = 2.3 andtherefore CMB at the time of decoupling naively should be consisted of about 104 causallydisconnected patches. However, we observe an almost perfectly uniform CMB temperaturefield across super-horizon scales at recombination. As we see in the following, cosmic inflation,an early period of accelerated expansion in which the SEC is violated, solves the horizonproblem dynamically and allows our universe to arise from generic initial conditions.

3.2 Cosmic inflation

The inflation paradigm postulates a brief period (within 10−34 s) of quasi-exponential accel-erated expansion during which the scale factor increased by over 60 e-folds. This considerableexpansion is sourced by a negative pressure component in energy-momentum of the mattercontents and drives the universe towards almost perfect homogeneity, isotropy, and flatnessthat we have observed.

Our discussion of horizon problem was based on the validity of SEC (1 + 3w > 0) andtherefore growing Hubble sphere of the standard Big Bang cosmology. A simple solutiontherefore is a phase of decreasing Hubble radius in the early history of the universe,

d(aH)−1

dt< 0 where 1 + 3w < 0. (77)

If this lasts long enough, e.g. aendainitial

= e60, the horizon problem can be solved.During this phase, the spacetime is very close to a de Sitter space and the Hubble pa-

rameter, H, is almost constant. The deviation from a perfect de Sitter space is quantified interms of two slow-roll parameters

ε ≡ − H

H2and η =

ε

Hε, (78)

which should be small during the slow-roll inflation. For a very nice lecture notes on differntaspects of cosmic inflation see The Physics of Inflation by Daniel Baumann.

4 Gravitational waves in cosmological eras

We perturb the metric around cosmological background and keep only the transverse trace-less perturbation, i.e. gravitational waves, as

ds2 = a2

(− dτ2 + (δij + γij)dx

idxj). (79)

16

Figure 7: Evolution of conformal horizon, (aH)−1 (red line), comparing with a given wavelength, k−1 (blueline), as a function of ln a. Image credit: Inflation by Daniel Baumann [10].

Inflation radiation matter

1

H

a

k

a

x

Figure 8: The Hubble radius, 1H

, and the physical wavelength, ak

, as functions of the scale factor. Notice thatX is the physcial coordinate, X = ax. The black line shows 1/H, the red and blue lines are the physicalwavelength of two modes which re-entered the cosmic Horizon during radiation domination (RD) and matterdomination (MD) eras respectively. The shaded yellow region shows causally connected points, while the grayshaded one shows our ignorance about reheating.

It is more convenient to go to the Fourier space and expand the field in terms of its polarizationstates

γij(τ,~k) =∑

σ=+,×eσij(k)γσ(τ,~k), (80)

17

where e+ij(k) and e×ij(k) are the plus and cross polarization tensors respectively. The field

equation of γσ(τ,~k) is

γ′′σ(~k) + 2Hγ′σ(~k) + k2γσ(~k) = 0, (81)

where again a prime denotes a derivative with respect to the conformal time andH = aH. Wecan extract analytically some general features of the solution by using the field redefinition

hσ(τ,~k) ≡ aγσ(τ,~k). (82)

The field equation takes the simple form

h′′σ(τ,~k) + (k2 − a′′

a)hσ(τ,~k) = 0, (83)

where ′ means derivative with respect to the conformal time and the effective mass terma′′

a = 12(aH)2(1− 3w) is

a′′

a=

2H2(1− ε) ' 2

τ2(1− ε) inflation

0 radiation-era12H

2 ' 2τ2

matter-era

. (84)

When the mode functions are outside the horizon (ka > 1), the gravitational wave is a constant

while when it is well inside the horizon (ka 1), it is a simple oscillating function scales like1/a

γσ(τ,~k) ∝

γ0 (ka 1)1a sin(ka + α) (ka 1)

, (85)

where γ0 and α (a phase), are both given by the initial conditions. See figure 9. The energydensity of the gravitational waves are given as

ρGW =1

32πG

∑σ=+,×

〈γ2σ〉, (86)

where the expectation value denotes average over several wavelengths. For gravitationalwaves inside the horizon, we have γ ∝ k cos(ka + α)/a2 which gives

ρGW ∝ a−4,

as it is expected from any form of radiation.

4.1 Inflation and primordial gravitational waves

Cosmic inflation generates primordial scalar perturbations that seeds all structure formationin the observable universe. More precisely, most of the inflationary models consistent withthe date predict an adiabatic and almost non-Gaussian scalar perturbation, the comovingcurvature perturbation ζ with a nearly scale invariant power spectrum as

∆2ζ = As(k∗)

( kk∗

)ns−1, (87)

18

Figure 9: The time evolution of γ+,×(τ, k), normalized to its initial value as a fucntion of x = log(a), forthree different values of the wavelength. Image credit: Gravitational Waves. Vol. 2: Astrophysics andCosmology [11].

which specifies in terms of two numbers As, and the spectral tilt

ns ≡ 1 +d lnPζd ln k

, (88)

which is of the order of slow-roll parameters. Another critical prediction of inflation is theexistence of a stochastic primordial GWs background (PGW) generated by tensor perturba-tions in the geometry of the very early universe. In this part, we discuss this mechanism.Just like CMB, this relic stochastic GWs is a random noise of GWs with no sharp, specificcharacters in either the time or frequency domains. However, GWs has an advantage overthe CMB because while photons decoupled about 4×105 years after the big bang, primordialGWs could free-stream from times as early as (possibly) Planck scales.

We are interested in metric perturbations that correspond at present time to gravitationalwaves. The linear Einstein equation for such a metric perturbation is given by Eq. (7) where∇µ is the covariant derivative in FLRW metric and it is sourced by the anisotropic stresstensor of matter fields

πij = δTij −1

3δijδT

kk . (89)

More precisely, we have

∂20γij + 3H∂0γij − a−2∂2

kγij = 8πGπTij , (90)

in which πTij is the traceless and transverse part of πij .4 Before going any further, note that

the tensor sector of perturbations, hij and πTij , are invariant under infinitesimal coordinatetransformations. Hence they are gauge invariant and physical quantities.

• Quantum origin of the perturbations:

4Using the standard scalar, vector, tensor decompostion, we can decompose πTij as

πij = ∂2ijπ

S − 1

3∂2πSδij + 2∂(iπ

Vj) + πTij ,

where ∂iπVi = ∂iπ

Tij = 0 [12]. Being a perfect fluid or having irrotational flows are physical properties,

thus their corresponding conditions are gauge-invariant. In other words, πS , πVi , πTij are all invariant under

infinitesimal space-time coordinate transformations.

19

For solving the field equation, we then need to set the initial condition. Assuming seeds ofperturbations to be from the quantum fluctuations, so-called Bunch-Davies vacuum, imposesthe initial value in the asymptotic past as

limτ→−∞

h+,×(τ,~k) =1√2ke−ikτ , (91)

hence both polarization states have the same initial values. In general relativity and the ab-sence of cosmological higher spin fields, e.g., gauge fields and fermions, to serve as a stochasticsource for gravitational waves, the anisotropic stress is zero. Therefore, the resulting pertur-bations are originated by vacuum fluctuations

h~k(τ) ≡ h+(τ,~k) = h×(τ,~k). (92)

which are un-polarized.

4.1.1 Adiabatic perturbations

Interestingly, regardless of the cosmological era which we have, in the limit that the physicalwavelength is much smaller than the Hubble rate, k/a H, the field equation (81) has twosolutions

h~k,1(τ) = cst. and h~k,2(τ) =

∫ τend

τ

dτ ′

a2(τ ′). (93)

which as we see, the second solution is decaying with time. Moreover, these are the solutions ofa second order differential equation so in the absence of any new degree of freedom, the aboveare all the possible super horizon solutions. Therefore at late times and for a genetic initialcondition, the gravitational waves eventually get dominated by h~k,1. The above solutions are

the adiabatic modes which are the result of the Weinberg’s adiabatic modes theorem [12].Here we only use its automatic result, but later in section 5.0.1, I will get back to it and discussthis theorem. The above simple observation has an amazing physical consequence. Recallingthat during inflation many Fourier modes leave the Hubble horizon which eventually duringradiation or matter era return to our causal patch, the above implies that once the adiabaticmode is outside the horizon it is conserved and constant, regardless of the complicated andunknown physics inside the horizon. Thus, this powerful effect allows us to connect thedistant past of our Universe to its recent past. Therefore, in the absence of the super-horizonentropy and anisotropic inertia perturbations, inflation predicts adiabatic fluctuations.

4.2 (Almost) scale invariant Power spectrum

Here by solving the field equation in (83). The general solution during inflation is expressedas a linear combination of Hankel functions

h~k(τ) '√π(k|τ |)

2ei(1+2ν

R)π/4

(c1H

(1)νR

(τ) + c2H(2)νR

(τ)), (94)

where

νR '3

2+ ε. (95)

20

Now, focusing on inflationary era and after imposing the usual Minkowski vacuum statefor the inflationary solution in (94), we obtain c1 = 1 and c2 = 0 in (94). The 2-point functionof gravitational waves is

〈γ~kγ~k′〉 =2π2

k3δ3(~k + ~k′)∆2

T (96)

where ∆2T is the power spectrum of the fluctuations. During inflation, the super-horizon

(k > aH) tensor power spectrum is

∆2T =

2

π2

H2

M2Pl

(k

aH)nT∣∣∣∣k=aH

, (97)

and its deviation from exact scale invariant, tensor spectral index nT , is

nT =d ln ∆2

T

d ln k= −2ε . (98)

The power spectrum predicted by inflation is specified by the energy scale of inflation and isnearly (but not exactly) scale invariant. Another important quantity is the tensor-to-scalarratio

r =∆T

∆ζ, (99)

which is the ratio of the power spectrum of GWs to the scalar curvature perturbations. Thecurrent upper limit on tensor fluctuations is

r0.05 < 0.07 at 95% CL

which comes from the latest joint analysis of Planck and BICEP2/Keck array measurements[13].

4.3 Nearly non-Gaussian stochastic field

At the level of linear approximation, inflationary fluctuations have a Gaussian probability.The statistical properties of an isotropic Gaussian fields are completely determined by the2-point function

〈ϕk1ϕk2〉 = (2π)3δ3(~k1 + ~k2)Pϕ(k1), (100)

while any odd-point function, e.g. its bispectrum, is exactly zero

〈ϕk1ϕk2ϕk3〉 = 0. (101)

This Gaussianity is a direct consequence of i) neglecting 2nd order terms in the equationof motion as well as ii) the cosmological principle. However, both of the above are notexact conditions during inflation and cosmic fluctuations break them both. As a result, thegenerated perturbations are not exactly Gaussian. The deviation from Gaussianity can beformulated in terms of the bispectrum as

〈ϕk1ϕk2ϕk3〉 = (2π)3δ3(~k1 + ~k2 + ~k3)Bϕ(k1, k2, k3). (102)

21

Figure 10: Marginalized joint 68 % and 95 % CL regions for ns and r at k = 0.002Mpc−1 from Planck aloneand in combination with BK14 or BK14 plus BAO data, compared to the theoretical predictions of selectedinflationary models. Image credit: Planck 2018 results X: Constraints on inflation [14].

If the power spectrum is scale-invariant, then the shape of the Bispectrum only depends ontwo numbers as

Bϕ(k1, k2, k3) = k−61 Bϕ(1, x2, x3), (103)

where x2 = k2k1

and x3 = k3k1

. In figure 12, we present the possible momentum configurationsof the bispectrum. Considering k1 to be not greater than k2 and k3, we have the followingthree possible limits. In the limit that k1 k2 ∼ k3, the bispectrum is called squeezed. Incase that k1 = k2 = k3 it is called equilateral and if k2 = k3, it is folded. Depends on thedetails of the inflationary model, the bispectrum has a peak in different limits.

Figure 11: The shaded area shows the possible momentum configurations of the bispectrum. Image credit:Primordial Non-Gaussianity by Daniel Baumann.

Computing the gravitons 3-point function as well as the combination of 2 scalars andone graviton Bispecturms in the squeezed limit for the vacuum GWs in general relativity,

22

one realizes that they are both slow-roll suppressed. The reason for that is the fact thatthe self-interaction terms are subleading by slow-roll parameters. In the squeezed limit, anadiabatic long (classical) wave gravitational wave acts as a coordinate transformation for theother short wavelength modes, either gravitons or scalars. That then leads to the Maldacena’spowerful consistency relation which we review in the next section.

5 Adiabatic modes and inflationary consistency relations

Here, we briefly review Weinberg’s adiabatic modes theorem and Maldacena’s inflationaryconsistency relations. For more details on Weinberg’s adiabatic modes see [12, 15] and forfurther information on Maldacena’s inflationary consistency relations see the seminal paper[16]. The adiabatic modes and consistency relation can be extended to include the gradientexpansions. Maldacena’s original consistency relation has been generalized to the conformalconsistency relation in [17] and to an infinite set of Ward identities in [18].

5.0.1 Weinberg’s Adiabatic modes

We start with the construction of Weinberg’s adiabatic modes in the Newtonian gauge whichis essential for the derivation of the consistency relations. Fixing the gauge, uniquely specifiesall the modes with finite momentum, k 6= 0. However, there are residual gauge symmetriesfor the very long-wavelength modes, which remain a symmetry of the gauge-fixed action.Starting with the flat FRW metric as the unperturbed metric

ds2 = −dt2 + a2dx2. (104)

Here we focus on the implications of the diffeomorphism invariance on the nature of the verylong-wavelength modes with k/H 1. Therefore, only the homogeneous diffeomorphismsare relevant. The general spatially homogeneous perturbed metric in the Newtonian gauge is

ds2 = −(1 + 2Φ(t))dt2 + a2

((1− 2Ψ(t))δij + γij(t)

)dxidxj , (105)

where Φ and Ψ are the Bardeen potentials and γij is the traceless tensor perturbation, gravitywave. 5 One can decompose the general spatially homogeneous perturbed energy-momentumin the Newtonian gauge as

T00 = −ρg00 + δρ(t) and T0i = −(ρ+ P )∂iδu(t), (106a)

Tij = P (t)gij(t) + a2

(δijδP (t) + ∂2

ijπS(t) + πTij(t)

), (106b)

where a bar denotes an unperturbed quantity, δρ, δP and δu are the perturbed density,pressure and velocity potential respectively. Moreover, πS and πTij are the scalar and tensor

anisotropic inertia which characterize departures of Tµν from the perfect fluid form. 6

Under the action of diffeomorphism transformations

xµ 7→ xµ = xµ + εµ(t,x), (107)

5Here we neglect the vector perturbations due to their damping nature in most of the inflationary modelsincluding our axion-gauge field model.

6It is interesting to note that in the decomposition (106b), the effects of bulk viscosity are included inδp [15].

23

there is a εµ which generates spatially homogeneous transformations on the metric and pre-serve the Newtonian gauge [12]

ε0(t,x) = −f(t)− χ(x), (108a)

εi(t,x) = (θδij + σij)xj − ∂iχ(x)

∫dt

a2(t), (108b)

where θ is a constant scalar and σij is a constant, traceless and symmetric matrix7, σii = 0.Therefore, choosing the Newtonian gauge, we are still left with residual gauge symmetriesfor the zero wavenumber modes. These diffeomorphisms do not vanish at spatial infinity andtherefore are called large gauge transformations. The scalar functions f(t), χ(t) and θ actonly on the scalar perturbations

Φ(t) 7→ Φ(t) + f(t) and Ψ(t) 7→ Ψ(t) + θ −Hf(t), (109)

while keep the tensor perturbations untouched. On the other hand, σij acts only on thegravitational waves as

γij(t) 7→ γij(t)− 2σij . (110)

Therefore, if Φ(t), Ψ(t) and γij(t) are solutions of the spatially homogeneous Einstein equa-tions, their transformed quantities and their differences are also the spatially homogeneoussolutions. In particular, in the scalar sector, we have spatially homogeneous solutions of theform

ΦA(t) = −f(t) and ΨA(t) = Hf(t)− θ, (111)

which corresponds to a cosmic fluid given as

δρA(t) = − ˙ρf(t), δPA(t) = − ˙Pf(t), δuA(t) = f(t), (112)

with a vanishing scalar anisotropy

πSA(t) = 0. (113)

That leads to a constant comoving curvature (R ≡ −Ψ + Hδu) and curvature perturbation(ζ ≡ −Ψ−Hδρ/ ˙ρ)

RA(t) = ζA(t) = θ. (114)

There is also a spatially homogeneous solution in the tensor sector as

γAij(t) = 2σij , (115)

with a vanishing tensor anisotropic inertia

πTAij (t) = 0. (116)

Up to now, the solutions (111) and (115) are only gauge degrees of freedoms for the k = 0mode and the Weinberg’s theorem relates them to the physical modes. The essential step in

7In general, εi(t,x) can have a constant term εi0 as well as a term like ωijxj where ωij = −ωji. Here, however,

we ignored them because (due to the spatial translational and rotational symmetry of the background metric)they do not have any effects on the linear perturbed metric.

24

Weinberg’s theorem is as follows. In case that the anisotropic inertia πij(t,k) and the entropy

perturbations,˙ρδP− ˙Pδρ3(ρ+P )2

, vanish in the limit k/H 1, the spatially homogeneous solutions

are extendible to modes with k 6= 0. Since solutions with non-zero wavenumbers have noresidual gauge symmetry in the Newtonian gauge, these modes are physical. When solvingthe linearized Einstein equations, there are two scalar and two tensor physical solutions whichare constant at k/H 1 (eq.s (114) and (115)), called adiabatic solutions.8 One immediateconsequence of this theorem is that these modes freeze out at horizon crossing and becomeindistinguishable from a redefinition of the background metric

gµν(t) + δgAµν(t) = ¯gµν(x), (117)

which implies

δgAµν(t) = −Lεgµν(t), (118)

where Lε denotes the Lie derivative with respect to εµ.

5.0.2 Gaussianity and Maldacena’s consistency relation

Now, we turn to the consistency relations which is a powerful probe of the early universephysics and holds under very general conditions, i.e., when the long-wavelength mode isadiabatic. Therefore, assuming Banch-Davis initial condition, the scalar consistency relationonly holds for single clock inflationary models in which the entropy perturbation is zero.However, gravity waves consistency relations hold for more general inflationary models. Moreprecisely, assuming Banch-Davis initial condition, only models in which πTij 6= 0 at super-horizon scales can violate tensor consistency relations, e.g., solid inflation [19] and anisotropicinflation [20]. As shown in [21], such inflationary models violate the cosmic no-hair conjecture.

Since the tensor modes are the main focus of this work, here we present the consistencyrelation for the gravity waves. The scalar consistency relation is the same and one only needsto replace γA

ij with the (adiabatic) curvature perturbation ζA. The key physical point behindthe consistency relations is the observation that the adiabatic long-wavelength modes can beremoved by the local coordinate transformation of the background metric, i.e. (118). Hence,they act as a classical background for the short wavelength modes, which freeze out muchlater than the long mode. In particular, an n-point correlation function of the short modescan be written as

〈ζ(x1)ζ(x2) · · · ζ(xn)〉γAij(x) = 〈ζ(x1)ζ(x2) · · · ζ(xn)〉, (119)

which Taylor expanding RHS around xi, we find the change of the short distance n-pointcorrelation function as

δ〈ζ(x1)ζ(x2) · · · ζ(xn)〉 =n∑

I=1

δ~xI.~∇I〈ζ(x1)ζ(x2) · · · ζ(xn)〉+ · · · , (120)

where xi and xi are related as

xi = xi +1

2γAijx

j . (121)

8This solution leads to equal values of δρx/ρx for all individual elements in the cosmic fluid which explainsthe name adiabatic.

25

As a result, the (n+1)-point correlation function including the long-wavelength mode is givenas

〈γAij(x)ζ(x1)ζ(x2) · · · ζ(xn)〉 ' 1

2

⟨γAij(x)γA

kl(x)n∑

I=1

xkI ∂l〈ζ(x1)ζ(x2) · · · ζ(xn)〉⟩, (122)

in which we only keep the dominate term that has the relevant contribution. The aboveequality is the consistency relation in real space. Going to the Fourier space, we can expandγij(q) as

γij(q) =∑λ=±

γλ(q)eij(q, λ), (123)

where eij(q, λ) are the time-independent polarization tensors, eij(q, λ)e∗ij(q, λ) = 2δλλ, inwhich λ = ± corresponding to the ±2 helicity states. Moreover, the two-point function atlate time is given as

〈γλ(q)γλ′(q)〉 = (2π)3δ(3)(q + q)P vacγ (q)δλλ

′, (124)

in which

P vacγ (q) = q−3

(H2

M2Pl

), (125)

is the power-spectrum. Then using the above and neglecting the gradients, we arrive at theMaldacena’s consistency relation

〈γλ(q)ζk1ζk2 · · · ζkn〉′ ' −1

2P vacγ (q)

n∑I=1

eij(q, λ)kIi∂kIj 〈ζk1ζk2 · · · ζkn〉′ for q → 0, (126)

where the prime in 〈· · · 〉 indicates that we extracted the prefactor (2π)3δ(3)(q +∑n

I kI)associated to momentum conservation. Note that the above result follows directly from thefact that adiabatic long-wavelength gravity wave is equivalent to a change of coordinate forthe short wavelength mode, regardless of the super-horizon behavior of the short modes.Therefore, as far as our inflationary system generates adiabatic tensor perturbations theabove consistency relation holds.

6 Gravitational waves and CMB

We now come to the effect of GWs on CMB anisotropies.

6.1 Integrated SachsWolfe effect

For that purpose we need to compute the effect of the perturbations on the photon trajectories

dPµ

dλ+ Γµνσ(xα(λ))P νP σ = 0, (127)

where Pµ = dxµ

dλ is the photon’s 4-momentum and Γµνσ(xα(λ)) is the Christoffel symbols upto the first order of perturbations. The photon’s energy measured with respect to the restframe of the baryon-photon fluid is

E(xν) ≡ −Pµ(xν)uµ(xν). (128)

26

Suppose that a photon emitted with energy EE at a point, xµE , and is observed at xµO withEO. Then, the temperature at xµE and at xµO in a rest frame of baryon-photon fluid are givenas

EOEE

=TOTE

. (129)

From the above and the photon’s geodesics (127), the temperature perturbation today indirection n is given in terms of the perturbations at the time of the photon decoupling, τdec,as

∆T (n)

T=

1

4δγ(τdec, rLn) + Ψ(τdec, rLn) + Φ(τdec, rLn)

+

∫ τ0

τdec

dτ

[Φ′(τ, (τ0 − τ)n) + Ψ′(τ, (τ0 − τ)n)− 1

2ninjh′ij(τ, (τ0 − τ)n)

],(130)

where rL = τ0− τdec, δγ(τdec, rLn) is the density perturbation in the radiation fluid and ni isthe propagation direction of the photon. The second line is the integrated SachsWolfe effectwhich receives contributions from the whole period from τrec to the present, τ0. That includesthe effect of a time-varying gravitational wave during the passage of the CMB photons fromthe last scattering to the present.

6.2 CMB polarization

Up to now, we considered the temperature fluctuations in CMB which its two-point temper-ature correlation function provides an important characterization of CMB anisotropies. Inaddition to the temperature anisotropies, there is more information to be gained by measur-ing CMB. More precisely, CMB photons are expected to be polarized due to the Thomsonscattering by free electrons before decoupling. The polarization of CMB photons providesthe cleanest and a promising method to detect primordial gravitational waves.

Figure 12: Left: A brief thermal history: nucleosynthesis, thermalization, recombination and reionization.Image credit: from Wyne Hu lecture notes [22]. CMB formed during the recombination epoch at redshiftzrec ' 1090, when free electrons became bound into hydrogen and helium. The universe back to full ionizationagain around z ∼ 10 leaving an extended neutral period between recombination and reionization. Right:Thomson scattering of CMB radiation by a free electron. An unpolarized radiation with quadratic anisotropybecomes linearly polarized by Thomson scattering.

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The polarization of an electromagnetic wave is described in terms of the Stokes param-eters. Consider a monochromatic electromagnetic plane wave that propagates along thedirection x3, can be decomposed as

~E = e−i(wt−~k.~x)

E1eiθ1

E2eiθ2

0

. (131)

We can describe the above radiation field with the 4 Stocks parameters

I ≡ 〈E21〉+ 〈E2

2〉, (132)

Q ≡ 〈E21〉 − 〈E2

2〉, (133)

U ≡ 2〈E1E2 cos(θ1 − θ2)〉, (134)

V ≡ 2〈E1E2 sin(θ1 − θ2)〉, (135)

where only 3 of the above are independent, i.e.

I2 = Q2 + U2 + V 2.

See figure 13 for the polarization corresponding to each of the Stokes parameters.

• The quantity I is total intensity of the wave, here the temperature anisotropy.

• Q describes the difference between the linear polarization in e1 and e2 directions.

• The U and V parameters give information on the phases. In particular, expanding thewave in the ±1 helicity polarization states, e± = (e1 ± ie2)/

√2, we have

V = 〈E+〉2 − 〈E−〉2, (136)

which represents the difference between the positive and negative helicity intensities.The U can be written as

U = 2〈E+E− sin(θ+ − θ−)〉. (137)

• Note that the mechanisms that generates CMB polarization produces only linear polar-izations, and no circular one. Therefore, in the absence of any parity violation interactionduring inflation, we can set V = 0.

• Finally, the angle of polarization is

tanα =E2

E1. (138)

Under a rotation of angle ϕ around the e3 as(x′

y′

)=

(cosϕ sinϕ− sinϕ cosϕ

)(xy

), (139)

the intensity, I, and the helical (circular) polarization, V , remain invariant. However, thelinear polarizations, Q, and U , transforms like spin-2 fields(

Q′(e3)± iU ′(e3))

= e±2iϕ(Q(e3)± iU(e3)

). (140)

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Therefore, I and V can be expanded in terms of scalar (spin-0) spherical harmonics, just asthe temperature is expanded as

Θ(n) =∑l,m

alm Ylm(n), (141)

while Q± iU can be expanded in terms of spin-weighted (spin-2) spherical harmonics as

Q(n)± iU(n) =∑l,m

a±lm ±2Ylm(n). (142)

For any arbitrary n direction, a function sf(n) defined on the sphere is called to be ofspin s if under a rotation of angle ϕ around the n, it transforms as

sf′(n) = e−isϕsf(n).

Any function of spin-s on the sphere can be expanded in terms of the spin-s weighted sphericalharmonics, sYlm(n). The spin-weighted spherical harmonics are related to the scalar (spin-0)spherical harmonics as

sYlm(n) ≡√

(l−s)!(l+s)!(∂

+)sYlm(n) for 0 ≤ s ≤ l (143)

sYlm(n) ≡√

(l+s)!(l−s)!(−1)s(∂−)−sYlm(n) for − l ≤ s ≤ 0, (144)

where ∂± are the spin raising and lowering operators given as

∂± sf(n) = − sin±s θ

(∂θ ±

i

sin θ∂φ

)[sin∓s θ sf(n)

]. (145)

Using (143) and (144), we can express the polarization as

(∂±)2(Q(n)± iU(n)

)=

√(l + 2)!

(l − 2)!

∑l,m

a±lm Ylm(n). (146)

Since the Stokes parameters, Q± iU , are not invariant under rotation, it is more convenientto expand them as (

Q(n)± iU(n))

=∑l,m

(aElm ± iaBlm

)±2Ylm(n). (147)

Therefore, one can define two scalar (spin-0) fields instead of the spin-2 fields, Q± iU , as

E(n) =∑l,m

aElmYlm(n) and B(n) =∑l,m

aBlmYlm(n). (148)

See left panel of figure 14.

• The scalar quantities E and B, completely determine the linear polarization fields.

• The E-mode is curl-free and even under the action of parity. Its polarization vectors areradial around cold sports (under dense) and tangential around hot spots (over dense).

• The B-mode is the divergence-free field and odd under parity. Its polarization vectorshave vorticity around the under and over dense areas.

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Figure 13: Left: The polarization corresponding to U = V = 0 (a and b), Q = V = 0 (c and d), and Q = U = 0(e and f). Right: E-mode and B-mode patterns of polarization. The polarization vectors around a cold spotand hot spot are shown in blue and red respectively.

Figure 14: E- and (delensed) B-mode power spectra for a tensor-to-scalar ratios r = 0.3, and for r = 0.01.Shown are also the experimental sensitivities for WMAP, Planck and two different realizations of a futureCMB satellite (CMBPol) (EPIC-LC and EPIC-2m).

• The angular power spectra are defined as

CXYl ≡ 1

2l + 1

∑m

〈aXlmaYlm〉 where X,Y = T,E,B. (149)

The autocorrelations of E- and B-modes, denoted by EE and BB, are presented in theright panel of figure 14.

• Scalar perturbations create only E-modes and no B-modes. However, vector perturba-

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tions create mainly B-modes.

• Tensor perturbations create both E-modes and B-modes.

Primordial gravitational waves and B-modes have not yet been detected. Recalling thatscalars do not produce B-modes while tensors do, a detection of primordial B-modes is asmoking gun of primordial gravitational waves, and therefore of inflation.

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